Download SIMNRA User's Guide - Max-Planck

MAX-PLANCK-INSTITUT FÜR PLASMAPHYSIK
GARCHING BEI MÜNCHEN
SIMNRA User’s Guide
Matej Mayer
IPP 9/113
April 1997
Die nachstehende Arbeit wurde im Rahmen des Vertrages zwischen dem
Max-Planck-Institut f¨
ur Plasmaphysik und der Europ¨
aischen Atomgemeinschaft u
¨ber die
Zusammenarbeit auf dem Gebiete der Plasmaphysik durchgef¨
uhrt.
IPP 9/113
Matej Mayer
SIMNRA User’s Guide
April 1997
Abstract
This report describes the use of the program SIMNRA and the physical concepts implemented therein. SIMNRA is a Microsoft Windows 95 / Windows NT program for the
simulation of back- or forward scattering spectra for ion beam analysis with MeV ions.
SIMNRA is mainly intended for the simulation of spectra with non-Rutherford backscattering cross-sections, nuclear reactions and elastic recoil detection analysis (ERDA). About
300 different non-Rutherford and nuclear reaction cross-sections for incident protons,
deuterons, 3 He and 4 He-ions are included. SIMNRA can calculate spectra for any iontarget combination including incident heavy ions and any geometry including arbitrary
foils in front of the detector. SIMNRA uses the Andersen-Ziegler values for the stopping
powers of swift and heavy ions and Chu energy loss straggling. Additionally SIMNRA can
calculate the effects of dual scattering. Data fitting (layer thicknesses, compositions etc.)
is possible by means of the Simplex algorithm.
This manual describes SIMNRA version 3.0
Contents
1 Overview
1
2 Installation
3
2.1
System requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3 Using SIMNRA
5
3.1
Basic steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
The File menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.3
The Edit menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.4
The Setup menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5
3.4.1
Setup:Experiment... . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4.2
Setup:Calculation... . . . . . . . . . . . . . . . . . . . . . . . . . 12
The Target menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5.1
Target:Target... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5.2
Target:Foil... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.6
The Reactions menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.7
The Calculate menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.7.1
Fit Spectrum... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8
The Plot menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.9
The Options menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.10 Adding new cross-section data
. . . . . . . . . . . . . . . . . . . . . . . . . 30
3.10.1 The R33 file format . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.11 Detector nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
i
4 Physics
34
4.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2
Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.1
Rutherford cross-sections . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.2
Non-Rutherford cross-sections . . . . . . . . . . . . . . . . . . . . . . 37
4.3
Evaluation of energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4
Stopping power data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5
4.6
4.4.1
Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4.2
Helium
4.4.3
Heavy ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.4
Stopping in compounds . . . . . . . . . . . . . . . . . . . . . . . . . 42
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Straggling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5.1
Electronic and nuclear energy loss straggling . . . . . . . . . . . . . 45
4.5.2
Energy loss straggling in compounds . . . . . . . . . . . . . . . . . . 52
Multiple and plural scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Examples
55
5.1
RBS: Rutherford cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2
RBS: Non-Rutherford cross-sections . . . . . . . . . . . . . . . . . . . . . . 58
5.3
ERDA: Non-Rutherford cross-sections . . . . . . . . . . . . . . . . . . . . . 60
ii
Chapter 1
Overview
This report describes the use of the program SIMNRA and the physical concepts implemented therein. SIMNRA is a Microsoft Windows 95 / Windows NT program for the
simulation of back- or forward scattering spectra for ion beam analysis with MeV ions.
SIMNRA is mainly intended for the simulation of spectra with non-Rutherford backscattering cross-sections, nuclear reactions and elastic recoil detection analysis (ERDA). About
300 different non-Rutherford and nuclear reaction cross-sections for incident protons,
deuterons, 3 He and 4 He-ions are included. SIMNRA can calculate spectra for any iontarget combination including incident heavy ions and any geometry including arbitrary
foils in front of the detector. SIMNRA uses the Andersen-Ziegler values for the stopping
powers of swift and heavy ions and Chu energy loss straggling. Additionally SIMNRA can
calculate the effects of dual scattering. Data fitting (layer thicknesses, compositions etc.)
is possible by means of the Simplex algorithm.
In contrast to other programs for the simulation of backscattering spectra SIMNRA is
easy to use due to the Microsoft Windows user interface. SIMNRA makes full use of the
graphics capacities of Windows.
This manual is organised in the following way:
• System requirements and the installation of the program are described in chapter 2.
• The use of the program is described in chapter 3. A quick overview about the steps
necessary to calculate a spectrum is given in chapter 3.1. More details are found in
the rest of chapter 3.
1
• The physical concepts implemented in the program are described in detail in chapter 4.
• Some examples for the abilities of the program are shown in chapter 5.
2
Chapter 2
Installation
2.1
System requirements
SIMNRA is a native 32-bit program for Windows 95 or Windows NT and requires at
least a 386-processor with math coprocessor or a higher processor (486, Pentium). A
fast processor (100 MHz 486 or higher) is highly recommended. SIMNRA will run under Windows 3.1 with WIN32S Version 1.30 or higher1 . However, using SIMNRA with
Windows 3.1 is not recommended and may result in global protection faults. SIMNRA
does not run under OS/2 Warp 3. Super-VGA resolution of 800 × 600 pixels or higher is
recommended. SIMNRA requires about 2 MB free hard disk space. At least 8 MB RAM
are recommended.
2.2
Installation
Unzip the file SIMNRAxy.ZIP2 into a directory of your choice, for example C:\SIMNRA.
Use pkunzip’s -d switch to create the correct subdirectory structure. After unzipping you
should have obtained the files listed in table 2.1.
Now run the program SIMNRA. The name of the executable program is SIMNRA.EXE.
When you run SIMNRA for the first time, you will be prompted for the name of the directory where SIMNRA is installed, for the above example enter C:\SIMNRA. The names
1
WIN32S
is
a
32-bit
extension
for
Windows
3.1.
It
is
available
from
Microsoft
(http://www.microsoft.com). WIN32S is not needed for Windows 95 or Windows NT. SIMNRA will
not run with WIN32S Version 1.25 or lower.
2
xy is the version number, 30 stands for version 3.0
3
Directory
\ATOM
\STOP
\CRSEC
\SAMPLES
Files
SIMNRA.EXE
README
MANUAL.PS
ATOMDATA.DAT
STOPH.DAT
STOPHE.DAT
LCORRHI.DAT
CHU CORR.DAT
CRSDA.DAT
*.R33
*.RTR
*.NRA
executable program
Readme file
This manual
atomic data
stopping data
cross-section data
examples
Table 2.1: Directory structure and files used by SIMNRA.
of the sub-directories will be created automatically.
Note: If you encounter error messages (File not found), then check the entries in
Options:Directories....
4
Chapter 3
Using SIMNRA
3.1
Basic steps
This section gives a quick overview about the basic steps necessary to calculate a backscattering spectrum.
Three steps must be performed before a backscattering spectrum can be calculated:
In a first step the experimental situation (incident ions, geometry) has to be defined, then
the target must be created, and in a third step the cross-sections used for the calculation
have to be chosen.
1. Click Setup:Experiment. Here you choose the incident ions, the ions energy, define
the scattering geometry (see fig. 3.2), and you enter the energy calibration of the
experiment.
2. Click Target:Target. Here you create the target. Each target consists of layers.
Each layer consists of different elements with some atomic concentration, which does
not change throughout the layer, and each layer has a thickness.
3. If there is a foil in front of the detector, then click Target:Foil for the definition
of a foil. The default is no foil in front of the detector. Like the target, the foil can
consist of different layers, and the layers can have different compositions.
4. Click Reactions. Here you have to choose which cross-section data should be used
for the simulation. The default are Rutherford cross-sections for all elements. You
can select non-Rutherford cross-sections instead and you can add nuclear reactions.
5
5. Now the spectrum can be calculated. Click Calculate:Calculate Spectrum for a
simulation of the spectrum.
6. With Setup:Calculation the parameters for the calculation can be altered. The
default values are normally sufficient, and you should change these values only if you
know what you are doing.
7. With File:Read data a measured spectrum can be imported for comparison with
the simulated one and for data fitting.
6
3.2
The File menu
In the File menu all necessary commands for reading and saving files and data, printing
spectra and terminating the program are located.
• New: This menu item resets the program to its starting values. All calculated spectra,
target, foil and setup definitions will be deleted.
• Open...: This menu item will read a saved calculation from disk.
• Save: This menu item will save all current parameters, target and foil definitions,
experimental and simulated data to disk. The data are saved as an ASCII file. The
default file extension is NRA.
• Save as...: Like Save, but you will be prompted for the name of the file.
• Read Data: This menu item allows the import of experimental data.
Read Data:IPP...: Reads experimental data stored in the data file format used at
the IPP Garching. This data file format will not be described here.
Read Data:ASCII...: Allows the import of experimental data in ASCII format.
The data file format must be as follows: The file may contain an arbitrary number
of comment lines at the beginning of the file. A comment line is a line that contains
any non-numeric character. These lines will be ignored. The first line that contains
only numeric characters will be treated as the first line of data. Each data line
must consist of two columns: In the first column the channel number must be given
(Integer), in the second column the number of counts must be given (Double). The
two columns are separated by an arbitrary number of blanks or tabs. Each line
must end with <CR><LF>1 . The data file may contain up to 8192 channels. An
example for a valid data file is given in fig. 3.1.
• Write Data...: This menu item exports the experimental and simulated data as
columns into an ASCII file. You can import this file easily into any plot program,
such as Excel, Origin or Mathematica.
1
<CR> means Carriage Return (#13 decimal), <LF> means Line Feed (#10 decimal).
7
This line
This line
Channel
1
2
3
4
<EOF>
may contain any comment <CR><LF>
may contain any comment as well <CR><LF>
Counts <CR><LF>
1000 <CR><LF>
1000.0 <CR><LF>
1.0E3 <CR><LF>
+1.0E3 <CR><LF>
Figure 3.1: Example for a valid data file which can be imported with File:Read Data:ASCII....
The first three lines will be ignored by the program. The channel number must be an integer
number, counts may be integer or floating point numbers.
The file format is as follows: The first line is a comment line which contains information about the contents of the different columns. The first column is the channel
number, the second column contains the experimental data (This column is set to
zero if experimental data are not available), the third column contains the simulated
data (This column is set to zero if simulated data are not available). If the Element
spectra option in Setup:Calculation... is checked, then the next columns will
contain the simulated spectra for each element in the target. The columns are separated with blanks.
• Print...: This menu item will print all parameters of the calculation and plot the
experimental and simulated curves.
Note 1: SIMNRA is not intended to produce high quality graphics, and the plot
looks quite ugly. If you want to obtain high quality graphics, you should use a
graphics program such as Excel or Origin. You can exchange data between SIMNRA
and any graphics program by file with File:Write Data... and via the clipboard
with Edit:Copy Data.
Note 2: Printing may not work properly if the program is used under Windows 3.1.
Printing works properly under Windows 95 and Windows NT.
• Exit: Terminates the program.
8
3.3
The Edit menu
• Copy Data: This will copy the experimental and simulated data in ASCII format to
the clipboard. They can be pasted into any spreadsheet program.
The format of the data in the clipboard is as follows: The data are organised in
three columns. The first column contains the channel number, the second column
contains the experimental data, and the third column contains the simulated data.
The columns are separated with tabs.
Note 1: SIMNRA does not support OLE.
Note 2: Copy Data does not work properly under Windows 3.1 and is disabled.
Copy Data works properly under Windows 95 and Windows NT.
9
3.4
The Setup menu
3.4.1
Setup:Experiment...
In the Setup:Experiment... menu the global parameters of the backscattering experiment are defined.
• Incident ion: Selects the incident ions. For incident protons (H), D, T, 3 He or
4 He
ions, the ions are selected by clicking the appropriate radio button.
For incident heavy ions select Other and enter the ions name in Other ion:Element
(for example Si, Cl, I). Lowercase and uppercase letters in the ions name are treated
similar, you can enter silicon as Si, si, SI or sI. The ions mass is selected from the
drop down box.
• Energy: Energy of the incident ions (in keV).
• Geometry: Geometry of the experiment: Incident angle α, exit angle β and scattering
angle θ. α and β are measured towards the surface normal, see fig 3.2. All angles in
degrees.
Note: 0◦ ≤ β < 90◦ . Transmission geometry is not possible.
• Calibration: Conversion from channels to energy. To account for detector nonlinearities, SIMNRA can use a non-linear energy calibration with a quadratic term of
the form
E [keV] = A + B × channel + C × channel2 .
(3.1)
E is the particle energy in keV. The calibration offset A must be entered in the
Calibration Offset field, A in keV. The energy per channel B must be entered
in the Energy per Channel field, B in keV/channel. C is the quadratic correction
term, C in keV/channel2 . For a linear energy calibration C = 0.0. A linear calibration is appropriate in most cases, and only if a high accuracy is intended a non-linear
calibration should be used.
Attention: If a non-linear energy calibration is used, then the energy scale, which
is plotted at the top axis, is only approximately valid: This scale remains linear even
for non-linear energy calibrations.
10
Figure 3.2: Geometry of a scattering experiment. Incident angle α, exit angle β and scattering
angle θ.
• Particles*sr: Number of incident particles times the solid angle of the detector.
Solid angle in steradians.
• Detector Resolution: Energy resolution of the detector (in keV). The energy resolution is measured as full width at half maximum (FWHM).
• Energy spread of incident beam: Usually the incident ion beam is not monoenergetic but has an energy distribution. SIMNRA assumes a gaussian energy distribution of the incident beam with a full width at half maximum which can be entered
in the Energy spread of incident beam field. The energy distribution of the incident beam depends on the experimental setup, typical values for this energy spread
are several keV in many experiments. If this field is set to 0.0 SIMNRA assumes a
monoenergetic incident beam.
11
3.4.2
Setup:Calculation...
In the Setup:Calculation... menu the parameters for the calculation can be altered.
This affects the accuracy of the calculation, but also the time necessary for the calculation
of a simulated spectrum.
• Stepwidth incoming ion: Stepwidth of the incoming ion (in keV). See chapter 4
for details. The default is 10 keV. The stepwidth of the incoming ion affects the
time T necessary to perform a calculation heavily. T depends on the stepwidth of
the incoming ion ∆E roughly as T ∝ 1/∆E: Decreasing the stepwidth by a factor
of two will roughly double the computing time.
For incident heavy ions with energies in the range of several ten MeV you can increase
this stepwidth to several 100 keV.
Note 1: The stepwidth of the incident ions is an important parameter for the accuracy of a simulation. If the stepwidth of the incident ion is too high, especially if
the exit angle β is close to 90◦ or the detector resolution is below 10 keV, unwanted
oscillations or steps in the simulated spectra may occur. This is due to rounding errors in the routine which calculates the contents of each channel. If these oscillations
occur you have to decrease the stepwidth of the incident ion.
Note 2: If the backscattering cross-section contains narrow resonances, the stepwidth of the incoming ions should be lower than the width of the resonance.
• Stepwidth Outgoing Particle: Stepwidth of outgoing particles (in keV). See chapter 4 for details. The default is 200 keV.
• Cutoff Energy: All particles are calculated until their energy has decreased below
the cut-off energy. You may speed up the calculation if you increase the cut-off
energy. The lowest possible value for the cut-off energy is 10 keV.
• Isotopes: If checked, backscattering from all isotopes of all elements in the target is
calculated. Especially for heavy elements with many isotopes this will slow down the
calculation significantly. If unchecked, the program will use only the mean masses
of the elements. Default is checked.
Important: Non-Rutherford cross-sections and nuclear reactions are only
available if Isotopes is checked.
12
• Straggling: If checked, electronic and nuclear energy loss straggling will be taken
into account. Default is checked.
Note: Straggling due to multiple small angle scattering is not calculated by the
program. See chapter 4 for details.
• High energy stopping: If checked, the program will use the high energy stopping formula by Andersen and Ziegler for incident protons and heavy ions for E >
1 MeV/amu.
If unchecked, the program will use the medium energy formula,
which is valid only in the range 10 keV/amu–1 MeV/amu, also at higher energies
E > 1 MeV/amu. The difference between the two formulas is very small in most
cases. This switch does not have any influence on the calculation of the stopping
power of helium ions and is disabled. For helium ions always the medium energy
formula is used, which is valid for all energies below 10 MeV. The default is checked.
Note: If you use high energy stopping, you may obtain kinks in the spectra because the two stopping power formulas do not fit absolutely smoothly together: The
derivative of the stopping power jumps at 1 MeV/amu.
• Dual Scattering: Most particles are scattered into the detector with only one
scattering event with large scattering angle. However, some particles may suffer
more than one scattering event with large scattering angle before they reach the
detector, see fig. 3.3. This is called plural scattering and results for example in
the low background behind the low energy edge of high Z layers on top of low Z
elements. The deviations at low energies between simulated and measured spectra
are also mainly due to plural scattering.
SIMNRA can calculate all trajectories with two scattering events. If Dual Scattering
is unchecked, then only one scattering event is calculated. This is the default. If
Dual Scattering is checked, additionally trajectories with two scattering events
will be calculated.
Warning: The calculation of dual scattering is a very time consuming process. If
Dual Scattering is checked, this will slow down the calculation of a spectrum by
a factor of about 200 (!), increasing the computing time from several seconds to at
least several minutes.
13
Figure 3.3: Examples of ion trajectories with one, two and three scattering events.
Note 1: Dual scattering should be used only if all cross-sections are Rutherford.
For the calculation of dual scattering the cross-sections for all possible scattering
angles between ≈ 0◦ and 180◦ must be known. This is only the case for Rutherford
cross-sections.
Note 2: SIMNRA calculates dual scattering only for incident ions and not for recoils
or reaction products of nuclear reactions. Additional scattering in a foil in front of
the detector (if any) is neglected.
Note 3: If Dual Scattering is checked, then Straggling must be checked too.
SIMNRA will check Straggling automatically, if Dual Scattering is checked. As
long as Dual Scattering is checked, Straggling cannot be unchecked.
• Element Spectra: If checked, individual spectra for each element in the target will
be calculated and plotted. If unchecked, only the total spectrum will be calculated
and plotted. Default is unchecked.
• Logfile: If checked, a file named SIMNRA.LOG will be created. This file contains
additional information about each step of the calculation. The logfile is intended for
debugging the program. Default is unchecked.
14
3.5
The Target menu
3.5.1
Target:Target...
In this menu the target is created. A target consists of layers. Each layer consists of
different elements and has some thickness. The composition of a layer does not change
throughout his thickness. To simulate a concentration profile you have to use multiple
layers.
The layer number 1 is at the surface of the target, the layer number 2 is below layer 1
and so on, see fig. 3.4.
• Thickness: Thickness of the layer (in 1015 atoms/cm2 ).
• Number of Elements: Number of different elements in this layer. The maximum
number of different elements in a layer is 20.
• Element: Name of the element, for example Si, W, Au. Lowercase and uppercase
letters in elements names are treated similar, you can enter silicon as Si, si, SI or sI.
XX means that this element is unknown. The special symbols D for deuterium, T
for tritium and A for 4 He can be used.
• Concentration: Atomic concentration of the element in the actual layer. The concentration c must be 0.0 ≤ c ≤ 1.0. The sum of the concentrations of all elements in
one layer must be equal to 1 (0.999 ≤
P
ci ≤ 1.001). If the sum of concentrations
is not equal to 1, the word concentration is written in red colour, if the sum of
concentrations is equal to 1, the word concentration is written in black colour. You
can use the small buttons to set the concentration of the element i to 1 minus the
sum of concentrations of all other elements, ci = 1 −
P
i6=j cj .
• Isotopes: You can use these buttons to change the concentrations of isotopes of that
element in the actual layer. You will need this only if this element does not have the
natural composition of isotopes. You can create for example a layer of enriched
on top of
12 C,
13 C
or the like. The sum of concentrations of all isotopes of one element
must be equal to 1.
Note: The Isotopes check-box in the Setup:Calculation... menu must be
checked to manipulate individual isotopes.
15
Figure 3.4: Layer structure of target and foil. For the target, layer 1 is at the surface, the layer
with the highest number is the deepest layer. Backscattered particles first penetrate the foil layer
with the highest number, the foil layer 1 is in front of the detector.
To manipulate layers use the buttons in the Layer manipulation box.
• Add: Adds a layer. The added layer will be the last layer. The maximum number of
different layers is 100.
• Ins: Inserts a layer in front of the current layer. The maximum number of different
layers is 100.
• Del: Deletes the current layer.
• Prev: Go to the previous layer.
• Next: Go to the next layer.
A layer can be copied to the clipboard with Edit:Copy Layer (or by pressing Ctrl C).
A layer can be pasted from the clipboard with Edit:Paste Layer (or by pressing Ctrl V).
Attention: If a layer is pasted, the current layer is overwritten.
16
3.5.2
Target:Foil...
In this menu a foil in front of the detector can be created. Like the target a foil can consist
of multiple layers with different compositions. See the previous section for details. If the
foil consists of multiple layers, then backscattered particles first will penetrate layer n,
then layer n − 1 etc., layer 1 is directly in front of the detector (see fig. 3.4).
The default is no foil in front of the detector.
17
3.6
The Reactions menu
In the Reactions menu the cross-sections used for the calculation of the simulated spectrum are chosen. Rutherford cross-sections for backscattering of projectiles and creation
of recoils are available for all ion-target combinations, if kinematically possible. Additionally SIMNRA can handle non-Rutherford cross-sections for backscattering and recoil
production and can use nuclear reactions cross-sections. SIMNRA is able to read three
different file formats with cross-section data:
1. The file CRSDA.DAT. This file was developed during the last years at the IPP
Garching and contains fitting coefficients to various cross-section data. A documentation is not available, and no guarantee is provided that the fits agree with the
original data. Data from this file should be used with care and only if no other data
are available.
2. The R33 file format. Cross-sections for nuclear reaction are stored in the R33 file
format in the SigmaBase data repository for ion beam analysis2 . Most files with
this extension have been taken from SigmaBase, a few ones were added by the author. The majority of these files has been digitised from the original publications by
G. Vizkelethy from Idaho State University. No guarantee is provided for agreement
with the original publication. The references of the original publications are found
in the file headers.
3. The RTR (Ratio To Rutherford) file format. These files contain non-Rutherford
cross-sections for backscattering of protons and α-particles. The files contain the
ratio of measured to Rutherford cross-sections. The majority of the data has been
digitised from the original publications by R.P. Cox, J.A. Leavitt and L.C. McIntyre, Jr. from Arizona University. These cross-section data have been published in
[1]. All files with this extension have been taken from SigmaBase. The references of
the original publications are found in [1].
SIMNRA distinguishes between three different types of scattering events for each isotope:
2
http://ibaserver.physics.isu.edu/sigmabase.
This
http://pixe.gns.cri.nz/.
18
server
is
mirrored
in
New
Zealand
at
1. Backscattering of projectiles
2. Creation of recoils
3. Nuclear reactions.
The chosen cross-sections for each type of scattering event must be unambiguous: You
can choose, for example, Rutherford cross-section for backscattering in the energy range
from 0.000–0.999 MeV, some non-Rutherford cross-section for backscattering in the energy
range from 1.000-1.999 MeV and a different cross-section for backscattering in the energy
range from 2.000–3.000 MeV. You cannot choose, however, Rutherford cross-section for
backscattering in the range 0.000–2.000 MeV and another cross-section for backscattering
in the energy range from 1.000–2.000 MeV: In this case the program does not know which
cross-section it should use in the range from 1.000–2.000 MeV, and you will get the error
message ’Energy overlap in cross-sections’.
All available cross-section data are listed in tables 3.1–3.3. If you want to add new
cross-section data files, see section 3.10.
Note 1: Files containing total cross-sections in the R33 file format are ignored. SIMNRA currently handles only differential cross-section data.
Note 2: Use the data files that came with SIMNRA. Some of the original data files
at SigmaBase contain small format errors, such as additional blank lines, which confuse
the program.
Note 3: Non-Rutherford cross-sections and nuclear reactions are only available if
Isotopes in the Setup:Calculation... menu is checked.
19
Table 3.1: Non-Rutherford backscattering cross-sections.
D(p,p)D
T(p,p)T
3
He(p,p)3 He
4
He(p,p)4 He
4
He(p,p)4 He
6
Li(p,p)6 Li
7
Li(p,p)7 Li
7
Li(p,p)7 Li
7
Li(p,p)7 Li
9
Be(p,p)9 Be
9
Be(p,p)9 Be
9
Be(p,p)9 Be
10
B(p,p)10 B
11
B(p,p)11 B
C(p,p)C
C(p,p)C
C(p,p)C
C(p,p)C
C(p,p)C
C(p,p)C
12
C(p,p)12 C
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
14
N(p,p)14 N
16
O(p,p)16 O
16
O(p,p)16 O
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
19
F(p,p)19 F
20
Ne(p,p)20 Ne
23
Na(p,p)23 Na
24
Mg(p,p)24 Mg
24
Mg(p,p)24 Mg
24
Mg(p,p)24 Mg
24
Mg(p,p)24 Mg
24
Mg(p,p)24 Mg
24
Mg(p,p)24 Mg
θ (Lab)
151
163.2
159.2
161.4
165
164
156.7
164
165
142.4
158.7
170.5
154
150
165
170
170
170
170
170
168.2
150
152
152
152
155.2
158.7
158.7
159.5
165
167.2
170
149.5
170
150
160
160
160
160
165
165
165
158.7
158.7
166 (CM)
156.5
164
164
164
164
164
164
Energy (keV)
1800–3000
2500–3500
2000–3000
1500–3700
1500–3000
1200–3100
373–1398
1700–3500
1300–2800
1600–3000
200–1700
2400–2700
1000–3000
500–2000
1000–3500
300–700
700–2800
300–3000
700–2500
996–3498
400–4500
800–1900
1035–1075
1450–1625
650–1800
1850–3000
1735–1760
1785–1815
600–4000
1850–3000
3600–4100
1450–2300
2450–2850
1000–3580
2000–5000
500–1300
1300–2064
500–1300
1300–1550
850–1010
1000–1875
1350–1550
600–1800
1300–1500
1500–2800
550–1450
400–4000
792–856
1466–1501
1642–1671
1991–2026
2393–2431
File
PH LA76A.RTR
PH LA76B.RTR
PHELA76A.RTR
PHELA76B.RTR
CRSDA.DAT, No. 5
PLIBA51A.RTR
PLIWA53A.RTR
PLIBA51B.RTR
PLIMA56A.RTR
PBEMO56A.RTR
PBEMO56B.RTR
PBELE94A.RTR
PB OV62A.RTR
PB TA56A.RTR
12CPPC.R33
PC LI93A.RTR
PC LI93B.RTR
PC LI93C.RTR
PC RA85A.RTR
PC AM93A.RTR
PC JA53A.RTR
PN TA56A.RTR
PN HA57A.RTR
PN HA57B.RTR
PN HA57E.RTR
PN LA67A.RTR
PN HA57C.RTR
PN HA57D.RTR
PN BA59A.RTR
PN LA67B.RTR
PN OL58A.RTR
PN RA85A.RTR
PO GO65A.RTR
PO AM93A.RTR
PF BO93A.RTR
PF DE56A.RTR
PF DE56B.RTR
PF DE56C.RTR
PF DE56D.RTR
PF KN89A.RTR
PF KN89B.RTR
PF KN89C.RTR
PF WE55A.RTR
PF WE55B.RTR
PNELA71A.RTR
PNABA56A.RTR
PMGMO51A.RTR
PMGMO51E.RTR
PMGMO51F.RTR
PMGMO51G.RTR
PMGMO51H.RTR
PMGMO51I.RTR
20
Reference
Langley 1976
Langley 1976
Langley 1976
Langley 1976
?
Bashkin 1951
Warters 1953
Bashkin 1951
Malmberg 1956
Mozer 1956
Mozer 1956
Leavitt 1994
Overley 1962
Trautfest 1956
Amirikas 1993
Liu 1993
Liu 1993
Liu 1993
Rauhala 1985
Amirikas 1993
Jackson 1953
Tautfest 1956
Hagedorn 1957
Hagedorn 1957
Hagedorn 1957
Lambert 1967
Hagedorn 1957
Hagedorn 1957
Bashkin 1959
Lambert 1967
Olness 1958
Rauhala 1985
Gomes 1965
Amirikas 1993
Bogdanovic 1993
Dearnaly 1956
Dearnaly 1956
Dearnaly 1956
Dearnaly 1956
Knox 1989
Knox 1989
Knox 1989
Webb 1955
Webb 1955
Lambert 1971
Bauman 1956
Mooring 1951
Mooring 1951
Mooring 1951
Mooring 1951
Mooring 1951
Mooring 1951
24
Mg(p,p)24 Mg
27
Al(p,p)27 Al
28
Si(p,p)28 Si
Si(p,p)Si
Si(p,p)Si
31
P(p,p)31 P
32
S(p,p)32 S
S(p,p)S
Cl(p,p)Cl
40
Ar(p,p)40 Ar
40
Ar(p,p)40 Ar
40
Ar(p,p)40 Ar
40
Ar(p,p)40 Ar
40
Ca(p,p)40 Ca
48
Ti(p,p)48 Ti
48
Ti(p,p)48 Ti
48
Ti(p,p)48 Ti
48
Ti(p,p)48 Ti
48
Ti(p,p)48 Ti
6
Li(α,α)6 Li
Li(α,α)7 Li
9
Be(α,α)9 Be
9
Be(α,α)9 Be
9
Be(α,α)9 Be
10
B(α,α)10 B
11
B(α,α)11 B
11
B(α,α)11 B
11
B(α,α)11 B
C(α,α)C
C(α,α)C
C(α,α)C
C(α,α)C
C(α,α)C
13
C(α,α)13 C
13
C(α,α)13 C
14
N(α,α)14 N
14
N(α,α)14 N
14
N(α,α)14 N
14
N(α,α)14 N
14
N(α,α)14 N
14
N(α,α)14 N
15
N(α,α)15 N
15
N(α,α)15 N
15
N(α,α)15 N
15
N(α,α)15 N
15
N(α,α)15 N
15
N(α,α)15 N
15
N(α,α)15 N
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
7
θ (Lab)
170
170
167.2
170
170
165
167.4
170
150
159.5
166 (CM)
166 (CM)
155
160
160
160
160
160
170
Energy (keV)
700–2540
1000–2450
1300–4000
1000–3580
1470–2200
1000–2000
1300–4000
1500–2690
2000–5000
1800–3600
1000–2000
1825–1950
1750–2750
1800–3000
1800–2150
2150–2500
2500–2800
2900–3040
1000–2600
File
PMGRA88A.RTR
PALRA89A.RTR
PSIVO59A.RTR
PSIAM93A.RTR
PSIRA85A.RTR
PP CO63A.RTR
PS OL58A.RTR
PS RA88A.RTR
PCLBO93A.RTR
PARBK61A.RTR
PARCO63A.RTR
PARCO63B.RTR
PARFR58A.RTR
PCAWI74A.RTR
PTIPR72A.RTR
PTIPR72B.RTR
PTIPR72C.RTR
PTIPR72D.RTR
PTIRA89A.RTR
Reference
Rauhala 1988
Rauhala 1989
Vorona 1959
Amirikas 1993
Rauhala 1985
Cohen-Ganouna 1963
Olness 1958
Rauhala 1988
Bogdanovic 1993
Barnhard 1961
Cohen-Ganouna 1963
Cohen-Ganouna 1963
Frier 1958
Wilson 1974
Prochnow 1972
Prochnow 1972
Prochnow 1972
Prochnow 1972
Rauhala 1989
112
121
136
157.5
170.5
170.5
150.8
160.5
170.5
149
165
167
170
170.5
165
165
163.7
165
167
167
167
167.2
165.2
165.2
165.2
165.2
165.2
165.2
165.2
158.6
165
165
165
2500–4500
2500–4500
6000–20000
1500–6000
575–4200
975–3275
2000–4000
4000–8000
980–3300
4000–12000
1810–9050
4000–7500
5000–9000
1560–5000
2000–3500
3300–6500
2600–4700
2000–6200
4550–6550
7090–9070
8650–9000
2000–4000
1600–2600
2400–3800
3800–4800
4700–5600
1600–5600
1600–5600
3800–4800
6000–10500
2050–9000
9200–9900
9600–10500
ALIBO72A.RTR
ALIBO72B.RTR
ABETA65A.RTR
ABEGO73A.RTR
ABELE94A.RTR
AB MC92A.RTR
AB RA72A.RTR
AB OT72A.RTR
AB MC92B.RTR
AC MS72A.RTR
AC FE94A.RTR
AC BI54A.RTR
AC CH94A.RTR
AC LE89A.RTR
AC BA65A.RTR
AC KE68A.RTR
AN KA58A.RTR
AN FE94A.RTR
AN FO93A.RTR
AN FO93B.RTR
AN FO93C.RTR
AN HE58A.RTR
AN SM61A.RTR
AN SM61B.RTR
AN SM61C.RTR
AN SM61D.RTR
AN SM61E.RTR
AN SM61F.RTR
AN MO72A.RTR
AO HU67A.RTR
AO FE94A.RTR
AO CA85A.RTR
AO CA85B.RTR
Bohlen 1972
Bohlen 1972
Taylor 1965
Goss 1973
Leavitt 1994
McIntyre 1992
Ramirez 1972
Ott 1972
McIntyre 1992
Marvin 1972
Feng 1994
Bittner 1954
Cheng 1994
Leavitt 1989
Barnes 1965
Kerr 1968
Kashy 1958
Feng 1994
Foster 1993
Foster 1993
Foster 1993
Herring 1958
Smothich 1961
Smothich 1961
Smothich 1961
Smothich 1961
Smothich 1961
Smothich 1961
Mo 1972
Hunt 1967
Feng 1994
Caskey 1985
Caskey 1985
21
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
16
O(α,α)16 O
18
O(α,α)18 O
F(α,α)F
19
F(α,α)19 F
19
F(α,α)19 F
19
F(α,α)19 F
Ne(α,α)Ne
Ne(α,α)Ne
Ne(α,α)Ne
Na(α,α)Na
Mg(α,α)Mg
24
Mg(α,α)24 Mg
24
Mg(α,α)24 Mg
24
Mg(α,α)24 Mg
24
Mg(α,α)24 Mg
24
Mg(α,α)24 Mg
Si(α,α)Si
Si(α,α)Si
28
Si(α,α)28 Si
28
Si(α,α)28 Si
28
Si(α,α)28 Si
28
Si(α,α)28 Si
27
Al(α,α)27 Al
Cl(α,α)Cl
Ar(α,α)Ar
39
K(α,α)39 K
Ca(α,α)Ca
40
Ca(α,α)40 Ca
θ (Lab)
165
165
165
165
165
165
165
165.7
170
170
160
165
170
170
170
167.3
167.3
167.3
165
165
162.5
162.5
164
165
165
170
170
165
165
165
165
170
165
170
175.5
166
145
Energy (keV)
10320–10700
10650–11100
11050–11600
11500–12500
12150–12750
12500–13500
9150–12750
5000–12500
2000–9000
1770–5000
2400–3500
1500–5000
1500–2300
2300–3700
1500–4000
2400–3200
3200–4000
2400–4000
2000–6000
2000–9000
3150–3900
4200–4900
3150–3900
5900–6250
6080–6140
2000–6000
6000–9000
2400–4000
4000–5000
5100–6000
2400–5000
2000–9000
2000–9000
1800–5200
6000–8000
2200–8800
5000–9000
File
AO CA85C.RTR
AO CA85D.RTR
AO CA85E.RTR
AO CA85F.RTR
AO CA85G.RTR
AO CA85H.RTR
AO CA85I.RTR
AO JO69A.RTR
AO CH93A.RTR
AO LE90A.RTR
AO PO64A.RTR
AF CH93A.RTR
AF CS84A.RTR
AF CS84B.RTR
AF CS84C.RTR
ANEGO54A.RTR
ANEGO54B.RTR
ANEGO54C.RTR
ANACH91A.RTR
AMGCH93A.RTR
AMGCS82A.RTR
AMGCS82B.RTR
AMGKA52A.RTR
AMGIK79A.RTR
AMGIK79B.RTR
ASICH93A.RTR
ASICH93B.RTR
ASILE72A.RTR
ASILE72B.RTR
ASILE72C.RTR
ASILE72D.RTR
AALCH93A.RTR
ACLCH93A.RTR
AARLE86A.RTR
AK FR82A.RTR
ACAHU90A.RTR
ACASE87A.RTR
22
Reference
Caskey 1985
Caskey 1985
Caskey 1985
Caskey 1985
Caskey 1985
Caskey 1985
Caskey 1985
John 1969
Cheng 1993
Leavitt 1990
Powers 1964
Cheng 1993
Cseh 1984
Cseh 1984
Cseh 1984
Goldberg 1954
Goldberg 1954
Goldberg 1954
Cheng 1991
Cheng 1993
Cseh 1982
Cseh 1982
Kaufmann 1952
Ikossi 1979
Ikossi 1979
Cheng 1993
Cheng 1993
Leung 1972
Leung 1972
Leung 1972
Leung 1972
Cheng 1993
Cheng 1993
Leavitt 1986
Frekers 1982
Hubbard 1990
Sellschop 1987
Table 3.2: Non-Rutherford ERDA cross-sections.
H(3 He,H)3 He
H(3 He,H)3 He
H(α,H)α
H(α,H)α
H(α,H)α
H(α,H)α
H(α,H)α
D(α,D)α
D(α,D)α
D(α,D)α
D(α,D)α
D(α,D)α
D(α,D)α
D(α,D)α
θ (Lab)
20
30
10
20
30
30
30
10
20
30
30
30
30
30
Energy (keV)
2000–3000
1900–3000
1000–2500
1000–2500
1000–2500
900–3000
2000–7000
1000–2500
1000–2500
1000–2500
2200–3000
1000–2070
2070–2180
2180–2800
File
1HTP1X1.R33
1HTP1X2.R33
ERD10H.R33
ERD20H.R33
ERD30H.R33
CRSDA.DAT, No. 125
CRSDA.DAT, No. 3
ERD10D.R33
ERD20D.R33
ERD30D.R33
CRSDA.DAT, No. 4
CRSDA.DAT, No. 130
CRSDA.DAT, No. 131
CRSDA.DAT, No. 132
23
Reference
Terwagne 1996
Terwagne 1996
Quillet 1994
Quillet 1994
Quillet 1994
Baglin 1991
?
Quillet 1994
Quillet 1994
Quillet 1994
?
Besenbacher 1986
Besenbacher 1986
Besenbacher 1986
Table 3.3: Nuclear reactions cross-sections.
D(3 He,α)p
D(3 He,α)p
D(3 He,α)p
D(3 He,p)α
3
He(D,α)p
3
He(D,p)α
6
Li(p,3 He)4 He
6
Li(p,α)3 He
6
Li(D,α)4 He
6
* Li(3 He,p0 )8 Be
*6 Li(3 He,p1 )8 Be
7
Li(p,α)4 He
9
Be(D,α0 )7 Li
9
Be(D,α1 )7 Li
9
Be(3 He,p0 )1 1B
9
Be(3 He,p1 )1 1B
9
Be(3 He,p0 )1 1B
9
Be(3 He,p1 )1 1B
10
B(p,α0 )7 Be
10
B(p,α1 )7 Be
10
B(p,α0 )7 Be
10
B(p,α1 )7 Be
10
B(D,α0 )8 Be
10
B(D,α1 )8 Be
10
B(3 He,p0 )12 C
10
B(3 He,p1 )12 C
10
B(α,p0 )13 C
10
B(α,p1 )13 C
11
B(3 He,D0 )12 C
11
B(3 He,p0 )13 C
11
B(3 He,p1,2,3 )13 C
12
C(D,p)13 C
12
C(D,p)13 C
12
C(3 He,p0 )14 N
12
C(3 He,p1 )14 N
12
C(3 He,p2 )14 N
13
C(D,p)14 C
14
N(D,α0 )12 C
14
N(D,α1 )12 C
14
* N(D,p0 )15 N
*14 N(D,p1,2 )15 N
*14 N(D,p3 )15 N
*14 N(D,p4,5 )15 N
*14 N(D,p5 )15 N
14
N(3 He,p1,2 )16 O
14
N(3 He,p1,2 )16 O
14
N(3 He,p3 )16 O
14
N(3 He,p3 )16 O
14
N(3 He,p4 )16 O
14
N(3 He,p4 )16 O
14
N(3 He,p5 )16 O
14
N(3 He,p5 )16 O
θ (Lab)
All
All
All
All
All
All
60
60
150
165
165
150
165
165
90
90
150
150
50
50
90
90
156
156
90
90
135
135
90
90
90
135
165
90
90
90
135
150
150
150
150
150
150
150
90
135
90
135
90
135
90
135
Energy (keV)
380–1000
700–2000
300–2000
380–1000
250–660
250–660
650–2900
650–2900
400–1900
900–5100
900–5100
500–1500
500–1900
500–1600
1800–5100
1800–5100
1800–5100
1800–5100
1800–10800
2350–10100
1800–9500
2650–7100
980–1800
980–1800
1300–5000
1300–5000
4000–5000
4000–5000
3000–5400
3000–5400
3000–5400
520–2950
800–1950
2100–2300
2100–2400
2100–2400
600–2950
600–1400
600–1400
500–1900
600–1400
800–1400
600–1400
600–1400
1600–2800
1600–2800
1600–2800
1600–2800
1600–2800
1600–2800
1600–2800
1600–2800
File
CRSDA.DAT, No. 1
CRSDA.DAT, No. 111
CRSDA.DAT, No. 129
CRSDA.DAT, No. 28
CRSDA.DAT, No. 2
CRSDA.DAT, No. 46
6LIP3HE.R33
6LIPA.R33
6LIDA 1.R33
6LITP0.R33
6LITP1.R33
7LIPA.R33
9BEDA0.R33
9BEDA1.R33
9BETP0 1.R33
9BETP1 1.R33
9BETP0 2.R33
9BETP1 2.R33
10BPA0 1.R33
10BPA1 1.R33
10BPA0 2.R33
10BPA1 2.R33
10BDA0.R33
10BDA1.R33
10BTP0.R33
10BTP1.R33
10BAP0.R33
10BAP1.R33
11BTD0.R33
11BTP0.R33
11BTP123.R33
12CDP 1.R33
12CDP 2.R33
12CTP0.R33
12CTP1.R33
12CTP2.R33
13CDP.R33
14NDA0 1.R33
14NDA1 1.R33
14NDP0 1.R33
14NDP12.R33
14NDP3.R33
14NDP45.R33
14NDP45.R33
14NTP1X1.R33
14NTP1X2.R33
14NTP3X1.R33
14NTP3X2.R33
14NTP4X1.R33
14NTP4X2.R33
14NTP5X1.R33
14NTP5X2.R33
24
Reference
?
?
?
?
?
?
Marion 1956
Marion 1956
Maurel 1981
Schiffer 1956
Schiffer 1956
Maurel
Biggerstaff 1962
Biggerstaff 1962
Wolicki
Wolicki
Wolicki
Wolicki
Jenkin 1964
Jenkin 1964
Jenkin 1964
Jenkin 1964
Purser 1963
Purser 1963
Schiffer 1956
Schiffer 1956
Giorginis 1995
Giorginis 1995
Holmgren 1959
Holmgren 1959
Holmgren 1959
Jarjis 1979
Kashy 1960
Tong 1990
Tong 1990
Tong 1990
Marion 1956
Amsel 1969
Amsel 1969
Simpson 1984
Amsel 1969
Amsel 1969
Amsel 1969
Amsel 1969
Terwagne 1994
Terwagne 1994
Terwagne 1994
Terwagne 1994
Terwagne 1994
Terwagne 1994
Terwagne 1994
Terwagne 1994
14
N(3 He,p7 )16 O
14
N(3 He,p7 )16 O
14
N(3 He,α0 )13 N
14
N(3 He,α0 )13 N
14
N(α,p0 )17 O
15
* N(p,α)12 C
15
N(D,α)13 C
16
O(D,α)14 N
16
O(D,α)14 N
16
O(D,α)14 N
16
O(D,p0 )17 O
16
O(D,p1 )17 O
16
O(D,p1 )17 O
16
O(3 He,α)15 O
18
* O(p,α)15 N
*18 O(p,α)15 N
18
O(D,α0 )16 N
18
O(D,α1 )16 N
18
O(D,α2 )16 N
18
O(D,α3 )16 N
19
* F(p,α)16 O
*19 F(p,α)16 O
19
F(D,α0 )17 O
19
F(D,α1 )17 O
θ (Lab)
90
135
90
135
135
140
150
135
145
165
135
135
155
90
155
165
165
165
165
165
90
150
150
150
Energy (keV)
1600–2800
1600–2800
1600–2800
1600–2800
4000–5000
900–2860
800–1300
800–2000
760–950
800–2000
20–3000
500–3000
400–1100
1600–2600
1500–1800
500–1000
830–2000
830–2000
830–2000
830–2000
700–1900
700–2000
700–1900
1000–1900
File
14NTP7X1.R33
14NTP7X2.R33
14NTA0X1.R33
14NTA0X2.R33
14NAP1.R33
15NPA.R33
15NDA 1.R33
16ODA 2.R33
16ODA 1.R33
16ODA 3.R33
16ODP0 1.R33
16ODP1 2.R33
16ODP1 1.R33
16OTA.R33
18OPA 2.R33
18OPA 1.R33
18ODA 1.R33
18ODA 2.R33
18ODA 3.R33
18ODA 4.R33
19FPA 1.R33
19FPA 2.R33
19FDA0 1.R33
19FDA1 1.R33
25
Reference
Terwagne 1994
Terwagne 1994
Terwagne 1994
Terwagne 1994
Giorginis 1995
Hagedorn 1957
Sawicki 1985
Amsel 1964
Turos 1973
Amsel 1964
Jarjis 1979
Jarjis 1979
Amsel 1967
Abel
Alkemada
Amsel 1967
Amsel 1964
Amsel 1964
Amsel 1964
Amsel 1964
Dieumegard 1980
Dieumegard 1980
Maurel 1981
Maurel 1981
3.7
The Calculate menu
In the Calculate menu all commands for calculating spectra, scattering kinematics, stopping powers and data fitting are located.
• Calculate Spectrum: Calculates the simulated spectrum.
• Calculate Spectrum Fast: Sets the detector resolution to 0.0 keV, ignores straggling and dual scattering and calculates the simulated spectrum. The calculation
of the spectrum is performed much faster than with Calculate Spectrum, but the
spectrum will contain kinks.
• Fit Spectrum...: Data fitting to experimental data. See section 3.7.1 for details.
• Kinematics...: Calculation of scattering kinematics. Allows the calculation of the
energies of backscattered particles, recoils and nuclear reaction products.
• Stopping...: Calculation of stopping powers for any projectile in any target element
and of energy loss in the different layers.
Note: The result of a stopping power calculation for E > 1 MeV/amu depends on
the setting of Setup Calculation:High energy stopping, see section 3.4.2.
3.7.1
Fit Spectrum...
Data fitting to backscattering spectra is a nontrivial task. In data fitting the quadratic
deviation of the simulated from the measured data points
χ2 =
X
w(i) (Nexp (i) − Nsim (i))2
(3.2)
i
is minimised by varying the input parameters of the calculation. Nexp (i) is the number
of counts in channel i of the measured spectrum, and Nsim (i) is the number of counts in
channel i of the simulated spectrum. w(i) is the weight of each data point.
Fast fitting algorithms, such as the Levenberg-Marquardt algorithm, tend to be unstable and require the knowledge of the derivatives of χ2 . SIMNRA uses the Simplex
algorithm for fitting [2]. The Simplex algorithm is very stable and converges (nearly)
always. However, the convergence is not very fast. The Simplex algorithm always uses
n + 1 points (called vertices) in the parameter space for fitting, where n is the number of
free parameters.
26
SIMNRA uses equal weighting of each data point, this means w(i) = 1 for all points.
You can fit:
1. Energy calibration
2. Particles*sr
3. Thickness of a layer
4. Composition of a layer
independently or all at once. Check which parameters should be varied. Only one layer
at a time can be fitted.
• Fitting range: The range (in channels) in which χ2 is calculated.
• Max Iterations: The maximum number of iterations. Fitting will be performed
until the desired accuracy is reached or the maximum number of iterations is reached.
• Max Error: The desired accuracy of the fit. The fit has converged if the relative
change of all fitted parameters and of χ2 is below Max Error. The relative change
of a parameter A is ∆A/A, where ∆A is the difference between the best vertex (the
vertex with the lowest χ2 ) and the worst vertex (the vertex with the highest χ2 ).
• fast and accurate fit: If fast fit is selected, the fit will be performed with zero
detector resolution and without calculation of straggling and dual scattering, see
Calculate Spectrum Fast. This will speed up the fit significantly, but is less accurate.
If accurate fit is selected, the actual detector resolution, straggling (if checked in the
Setup:Calculation menu) and dual scattering (if checked in the Setup:Calculation
menu) are used for the calculation.
27
3.8
The Plot menu
This section describes all plot related commands, including all commands which are not
accessible via menus.
• Autoscaling: If checked, the plot will be scaled automatically to minimum and
maximum if experimental data are imported or a new calculation is performed. If
unchecked, the axis scales remain fixed.
• Delete experimental data: Deletes the experimental data from the plot.
• Delete simulated data: Deletes all simulated data from the plot.
• Scaling the axis: To scale the x- or y-axis double-click with the left mouse button
on the desired axis. Enter the axis minimum and maximum.
• Zooming into the plot: To zoom into the plot click with the left mouse button
into the upper left corner of the range you want to zoom in. Keep the mouse button
down and tear a rectangle to the lower right corner of the zooming range. Release the
left mouse button. Now double-click with the left mouse button into the rectangle
to zoom in.
• Zooming out: Click the right mouse button to zoom out.
28
3.9
The Options menu
• Directories: Directories where atomic data (file ATOMDATA.DAT), stopping data
(the files STOPH.DAT and STOPHE.DAT) and cross-section data are located.
• Create Reaction List: SIMNRA uses a file named CRSEC.LST in the crosssections directory to know which cross-section data are available. Create Reaction
List will create this file. You have to recreate the reaction list if you add or delete
cross-section data files.
Note: Some data files contain total cross-sections. These files are ignored by SIMNRA. The program displays a list of all ignored files.
29
3.10
Adding new cross-section data
To add new cross-section data, you have to perform the following steps:
1. Create a cross-section data file in the R33 file format. The file format is described
below.
2. Copy this file into the directory where all other cross-section data files are. This
directory is displayed in Options:Directories....
3. Recreate the reaction list by clicking Options:Create Reaction List.
Note: If your file will be ignored, then SIMNRA was not able to read or understand
the file. Carefully read the section about the R33 file format and try again.
3.10.1
The R33 file format
The R33 file format is described in full detail in a proposal by I.C. Vickridge. The proposal
can be obtained from SigmaBase. An example for a valid file in the R33 format is shown
in fig. 3.5. Each line must end with <CR><LF> (Carriage Return and Line Feed).
SIMNRA uses not only the data points, but also a part of the information supplied in
the file header. The following lines must be present in the file in the given order. The file
may additionally contain an arbitrary number of other information. All other lines than
the ones listed below are ignored by the program.
• A line containing the string ’REACTION:’. Uppercase characters are important.
SIMNRA will interpret the nuclear reaction string written in that line (In the example of fig. 3.5 16O(d,a0)14N) to find out which particles are involved in the nuclear
reaction. The masses of the particles are ignored.
• A line containing the string ’MASSES:’. SIMNRA will read the masses of the particles from this line. Please note that the first mass is the mass of the incident particle,
the second mass is the mass of the target particle, the third mass is the mass of the
outgoing particle for which the cross-section is valid and the fourth mass is the mass
of the other reaction product.
• A line containing the string ’QVALUE:’. The Q-value is the energy released in the
nuclear reaction (in keV).
30
COMMENT: These cross sections have been digitised from the publication
cited below. No error of either the energy and or the sigma is given.
Some errors may be among the data, we are recently checking them.
The Los Alamos Ion Beam Handbook also will contain these data as soon
as it is ready. If you use this data please refer to the paper below.
Source: A.Turos, L.Wielunski and a Batcz, NIM, 111(1973), 605
Special comment:
WARNING ! THIS IS MAINLY FOR TEST NO GUARANTY IS PROVIDED FOR EVEN
AGREEMENT WITH THE ORIGINAL PUBLICATION.
NAME: Gyorgy Vizkelethy
ADDRESS1: Department of Physics
ADDRESS2: Idaho State University
ADDRESS3: Campus Box 8106
ADDRESS4: Pocatello, ID 83209-8106
ADDRESS5: (208) 236-2626
ADDRESS6: [email protected]
SERIAL NUMBER: ?
REACTION: 16O(d,a0)14N
DISTRIBUTION: Energy
MASSES: 2,16,4,14
QVALUE: 3110.00
THETA: 145.0
SIGFACTORS: 1.00 0.00
ENFACTORS 1.00 0.00 0.00 0.00
NVALUES: 5
761.0
0.0
2.92E+0000
0.0
770.0
0.0
3.65E+0000
0.0
775.0
0.0
3.99E+0000
0.0
780.0
0.0
4.41E+0000
0.0
785.0
0.0
4.55E+0000
0.0
Figure 3.5: Example for a cross-section data file in the R33 file format.
31
• A line containing the string ’THETA:’. Theta should be given in degrees. The value
of theta is not used by SIMNRA, however this line must be present and contain a
value.
• A line containing the string ’NVALUES:’. The value of nvalues is ignored, however
this line must be present. SIMNRA assumes that the data will start after this line.
• The data are organised in 4 columns: The first column is the energy in keV, the
second column is the energy error (ignored by SIMNRA), the third column is the
differential cross-section in the laboratory frame measured in mbarn/sr, and the
fourth column is the cross-section error (ignored by SIMNRA). SIMNRA expects
the data to be arranged in order of ascending energy.
32
3.11
Detector nonlinearity
It is well known that the energy calibration of semiconductor detectors, which are used
in most ion beam experiments, is not exactly linear. This is due to the energy dependent
energy loss in the top electrode and the dead layer of the semiconductor detector [3, 4],
which results in a nonlinearity of typically several channels.
To account for detector nonlinearities, SIMNRA offers two possibilities:
1. You can create a foil consisting of two layers in front of the detector. Layer 2 is
composed of the material of the top electrode (usually Au) and has the thickness of
the electrode (usually the thickness is supplied by the manufacturer of the detector).
Layer 1 is composed of silicon in the case of a silicon detector and has the thickness
of the dead layer (the dead layer is the insensitive region near the electrode). The
dead layer thickness can be obtained only experimentally by tilting the detector.
2. SIMNRA offers the possibility to use a non-linear energy calibration with a quadratic
correction term of the form
E [keV] = A + B × channel + C × channel2 .
See section 3.4.1 for details.
33
Chapter 4
Physics
4.1
Overview
During the last decade, several programs for the simulation of backscattering spectra have
been developed. The most common program is Doolittle’s RUMP [5, 6]. However, RUMP
uses many approximations to save computing time. The increase in computer power during
the last years has made it possible to drop several of the approximations used by RUMP.
SIMNRA offers more freedom in the use of non-Rutherford cross-sections and nuclear
reactions, treats several topics such as straggling and convolution more precise and adds
new possibilities such as dual scattering. This section describes the physics involved in
the simulation of a backscattering spectrum as performed by SIMNRA.
The target is subdivided into shallow sublayers. Each simulated spectrum is made
up of the superimposed contributions from each isotope of each sublayer of the sample
target. The thickness of each sublayer is chosen in such a way that the energy loss in each
sublayer is about the stepwidth of the incoming particles. When the incident particles
penetrate a sublayer, they loose energy due to electronic and nuclear energy loss and the
beam energy is spread due to straggling. The calculation of the energy loss is described in
detail in section 4.3, and the calculation of straggling in section 4.5. SIMNRA calculates
the energy of backscattered particles1 from the front and the backside of the sublayer, and
the energy of these particles when reaching the detector after passing to the target surface
and traversing a foil in front of the detector, see fig. 4.1. The contribution of each isotope
in each sublayer will be referred to as a brick.
1
A ‘backscattered’ particle may be a recoil or a product in a nuclear reaction as well.
34
Figure 4.1: Notation used for a single brick.
To account for energy straggling and the finite energy resolution of the detector the
brick shown in fig. 4.1 is convoluted with a Gaussian function f (E, σ 2 ) with width
2
2
σ 2 = σStraggling
Out + σDetector .
(4.1)
2
σStraggling
Out is the variance of the energy distribution of the outgoing particles due to
2
energy loss straggling, and σDetector
is the energy resolution of the detector.
The final contribution to the energy spectrum of each isotope in each sublayer is given
by
S(E) =
Z ∞
0
S0 (E 0 ) f (E 0 , σ 2 (E 0 )) dE 0
(4.2)
Here S0 (E) is the energy spectrum before convolution and S(E) the spectrum after the
convolution. Note that the width of the Gaussian changes throughout the brick due to
different straggling contributions.
The Number of counts Ni in each channel i is given by integrating S(E) over the
channel width from the minimum to the maximum energy of each channel:
Ni =
Z Emax (i)
Emin (i)
S(E 0 ) dE 0
(4.3)
Eqs. 4.2 and 4.3 can be put together into a 2-dimensional integral, which is computed by
SIMNRA by means of a 2-dimensional Gauss-Legendre integration. The accuracy of the
integration is about 10−4 .
The area Q of the brick in fig. 4.1 is calculated by SIMNRA by using the cross-section
35
¯ in the brick.
at the mean energy E
Q = N ∆Ω
dσ ¯ ∆x
(E)
dΩ
cos α
(4.4)
¯
N ∆Ω is the number of incident particles times the solid angle of the detector and dσ/dΩ(E)
¯ The heights of the front
is the differential cross-section evaluated at the mean energy E.
and back side of the brick are adjusted to give the correct area when integrated. SIMNRA
interpolates the brick linearly, as shown in fig. 4.1. This is, however, only valid if the
cross-section does not vary strongly and the brick is sufficiently thin. If the cross-section
has structures such as sharp resonances, the stepwidth of the incoming particles must be
sufficiently smaller than the width of the resonance.
4.2
Cross-sections
4.2.1
Rutherford cross-sections
The Rutherford cross-section for backscattering is given in the laboratory system by
µ
σR [mb/sr] = 5.18275 × 106
Z1 Z2
E [keV]
¶2
n¡
M22 − M12 sin2 θ
¢1/2
+ M2 cos θ
¡
M2 sin4 θ M22 − M12 sin2 θ
¢1/2
o2
(4.5)
θ is the scattering angle, Z1 and M1 are the nuclear charge and the mass of the projectile,
respectively, and Z2 and M2 are the nuclear charge and the mass of the target atom,
respectively. σR is the differential cross-section in the laboratory system. Experimental
measurements indicate that actual cross-sections deviate from Rutherford at both high
and low energies for all projectile-target pairs. The low-energy departures are caused by
partial screening of the nuclear charges by the electron shells surrounding both nuclei
[7, 8, 9, 1]. This screening is taken into account by a correction factor F :
σ = F σR
For θ > 90◦ the correction factor by L’Ecuyer et al. [7] is widely used:
4/3
0.049 Z1 Z2
FL’Ecuyer = 1 −
ECM
(4.6)
ECM is the energy in the center of mass system (in keV). Tabulated values of FL’Ecuyer
can be found for example in [1]. The correction for backscattering angles θ > 90◦ at
36
typical energies used in ion beam analysis usually is small. For 1 MeV 4 He ions on gold
the correction is only about 3.5%.
The correction factor by L’Ecuyer (eq. 4.6) is a first order correction and does not take
into account the influence of the scattering angle θ. For θ < 90◦ eq. 4.6 will underestimate
the necessary correction to the Rutherford cross-section. SIMNRA uses the angular- and
energy dependent correction factor by Andersen et al. [9]:
³
1+
FAndersen = ½
1+
V1
ECM
h
+
1 V1
2 ECM
2ECM
´2
V1
sin θCM /2
i 2 ¾2
(4.7)
θCM is the scattering angle in the center of mass system. The increase in the kinetic
energy V1 is given by
³
2/3
V1 [keV] = 0.04873 Z1 Z2 Z1
´
2/3 1/2
+ Z2
The dependence of the correction factor FAndersen from the scattering angle θ for 4 He
scattered from gold is shown in fig. 4.2 for different 4 He energies. Dashed lines are the
angular independent correction factor by L’Ecuyer. For large scattering angles the correction factors by L’Ecuyer and Andersen are near to unity and similar, however, for small
scattering angles the correction by Andersen becomes large and the angle-independent
L’Ecuyer correction underestimates the deviations from the Rutherford cross-section.
The Rutherford cross-section for recoils is given in the laboratory system by
ERD
σR
[mb/sr] = 2.0731 × 107
[Z1 Z2 (M1 + M2 )]2
(2M2 E [keV])2 cos3 θ
(4.8)
θ is the recoil angle in the lab system. SIMNRA applies the correction to the Rutherford
cross-section from eq. 4.7 also for the recoil cross-section.
4.2.2
Non-Rutherford cross-sections
At high energies the cross-sections deviate from Rutherford due to the influence of the
nuclear force. A useful formula above which energy EN R deviations from Rutherford can
be expected was given by Bozoian [10, 11, 12]:
EN R [MeV] =
EN R [MeV] =
M1 + M2 Z2
for Z1 = 1
M2
10
M1 + M2 Z1 Z2
for Z1 > 1
M2
8
37
1.00
2000 keV
1000 keV
0.95
500 keV
0.90
250 keV
F(E, θ)
0.85
0.80
0.75
0.70
0.65
0.60
0
30
60
90
120
150
180
θ (degree)
Figure 4.2: Angular dependence of the correction factors for the Rutherford cross-section by
L’Ecuyer (eq. 4.6, dashed lines) and Andersen (eq. 4.7, solid lines) for 4 He backscattered from gold
at different energies.
EN R is the energy at which the deviation from the Rutherford cross-section is > 4%.
SIMNRA does not check if the cross-sections at a given energy are Rutherford or not. It is in the responsibility of the user to choose the correct crosssections. The above formulas may be useful to estimate if the cross-section is
still Rutherford or not.
For non-Rutherford cross-sections SIMNRA uses experimentally determined differential cross-sections taken from SigmaBase. The use of non-Rutherford cross-sections is
described in full detail in section 3.6. SIMNRA uses linear interpolation between the
given data points.
4.3
Evaluation of energy loss
The energy E of a particle in the depth x is given by the integral equation
E(x) = E0 −
Z x/ cos α
dE
dx0
0
(E(x0 ), x0 )dx0
(4.9)
Here we assume that the particle starts with initial energy E0 at the surface (x = 0),
dE/dx0 (E(x0 ), x0 ) is the energy and depth dependent stopping power. In principle, eq. 4.9
38
can be evaluated directly, however this consumes a lot of computing time.
For the evaluation of the energy loss SIMNRA uses the algorithm of Doolittle instead,
developed for RUMP [5]. The beam loses energy according to the differential equation
dE
= −²(E)
dx
(4.10)
where this is the defining equation for ²(E), the energy dependent stopping cross-section. x
is the pathlength into the material, measured in areal density (1015 atoms/cm2 ). SIMNRA
uses the Ziegler stopping power data, see section 4.4. ²0 = d²/dE is the first and ²00 =
d2 ²/dE 2 the second derivative of ².
If incoming or outgoing particles with incident energy E0 traverse a layer of material
with thickness ∆x, then the particles energy E1 after the layer can be expanded into a
Taylor series:
E1 = E0 + ∆x
dE
1
d2 E
d3 E
1
(E0 ) + ∆x2 2 (E0 ) + ∆x3 3 (E0 )
dx
2
dx
6
dx
(4.11)
The terms in eq. 4.11 look as follows:
dE
dx
d2 E
dx2
d3 E
dx3
= −²
=
=
(4.12)
d² dE
d
(−²) = −
= ²0 ²
dx
dE dx
d²0
d²
d 0
(² ²) =
² + ²0
= −²00 ²2 − ²02 ²
dx
dx
dx
(4.13)
(4.14)
With ², ²0 and ²00 evaluated at E0
³
´
1
1
E1 = E0 − ∆x² + ∆x2 ²²0 − ∆x3 ²00 ²2 + ²02 ²
2
6
(4.15)
See ref. [5] for a discussion of the accuracy of the above equations.
²0 and ²00 are calculated by SIMNRA by numerical differentiation of the Ziegler stopping
power data. The stepwidth ∆x for the incoming and outgoing particles can be adjusted
in the Setup:Calculation menu. The stepwidth of the incoming particle should be kept
small in the range of 10 keV due to the cross-section calculation, see section 4.1. The
stepwidth for outgoing particles can be chosen much larger due to the accuracy of eq. 4.15.
Typical values for the stepwidth of outgoing particles are around 200 keV.
39
4.4
4.4.1
Stopping power data
Hydrogen
SIMNRA uses the electronic stopping power data by Andersen and Ziegler [13] for the
stopping of incident protons, deuterons and tritons in all elements. The electronic stopping
power Se in eV/(1015 atoms/cm2 ) for an incident hydrogen ion with energy/mass E in
keV/amu is given by
1
1
1
=
+
Se
SLow
SHigh
(4.16)
SLow = A2 E 0.45
(4.17)
with
and
·
SHigh
A3
A4
=
ln 1 +
+ A5 E
E
E
¸
(4.18)
A2 −A5 are fitting coefficients and tabulated in [13]. They are stored in the file STOPH.DAT.
Equations 4.16–4.18 are valid for 10 keV ≤ E < 1 MeV. For energies in the range 1 MeV–
100 MeV the electronic stopping power Se is given by
"
4
X
A6
A7 β 2
2
Se = 2 ln
−
β
−
Ai+8 (ln E)i
β
1 − β2
i=0
#
(4.19)
A6 − A12 are tabulated in [13], β = v/c, with v the ion velocity and c the speed of light.
Equation 4.19 is used only if the switch High energy stopping in the Setup:Calculation
menu is checked. If unchecked, the program will use Equations 4.16–4.18 at all energies.
The program default is checked. The difference between eq. 4.16 and eq. 4.19 is small in
most cases. The main problem using eq. 4.19 is that the first and second derivatives of
eq. 4.16 and eq. 4.19 do not fit smoothly together at 1 MeV/amu. This may result in the
appearance of kinks in the spectrum.
Nuclear stopping for incident hydrogen, deuterium and tritium ions is negligible for
incident energies above about 10 keV/amu [13] and is neglected by SIMNRA.
4.4.2
Helium
For incident 3 He and 4 He ions the electronic stopping power data by Ziegler [14] are used
for all elements. The electronic stopping power Se in eV/(1015 atoms/cm2 ) for incident
40
4 He
ions with energy E in keV is given by
1
1
1
+
=
Se
SLow
SHigh
(4.20)
SLow = A1 E A2
(4.21)
with
and
·
SHigh
A3
A4
=
ln 1 +
+ A5 E
E
E
¸
(4.22)
A1 −A5 are fitting coefficients and tabulated in [14]. They are stored in the file STOPHE.DAT.
Equations 4.20–4.22 are valid for 1 keV ≤ E < 10 MeV. 4 He-stopping at higher energies
(above 10 MeV) is not implemented in the program.
The stopping power of 3 He is identical to the stopping power of 4 He at the same
velocity [14]. The stopping power of 3 He with energy E is obtained by taking the stopping
power value at the energy E(4 He) = 4/3 E(3 He). The stopping power for 3 He is valid in
the energy range 10 keV ≤ E < 7.5 MeV.
Nuclear stopping for incident helium ions is calculated with the Krypton-Carbon (KrC) potential [15]. The nuclear stopping Sn in eV/1015 atoms/cm2 for He-ions with incident
energy E (in keV) is given by:
8.462 Z1 Z2 M1
Sn = sn
³
2/3
(M1 + M2 ) Z1
´
2/3 1/2
(4.23)
+ Z2
sn is the reduced nuclear stopping and Z1 , M1 are the nuclear charge and mass of the
helium ion and Z2 , M2 are the nuclear charge and mass of the target element. The reduced
nuclear stopping sn has the simple form
sn = 0.5
ln(1 + ²)
² + 0.10718 ²0.37544
(4.24)
² is the reduced energy and is given by
32.53 M2 E
²=
³
2/3
Z1 Z2 (M1 + M2 ) Z1
´
2/3 1/2
(4.25)
+ Z2
Nuclear stopping is only important at incident energies E < 100 keV, at higher energies
nuclear stopping becomes negligible.
41
4.4.3
Heavy ions
The electronic stopping power of heavy ions in all elements is derived from the stopping
power of protons using Brandt-Kitagawa theory [16]. The formalism is described in detail
in ref. [16]. The screening length Λ (eq. 3-29 of ref. [16]) is multiplied by an empirical
correction factor which has been digitised from fig. 3-25 of ref. [16]. The correction factor
for all elements is stored in the file LCORRHI.DAT.
Note that the switch High energy stopping in the Setup:Calculation menu has
influence on the calculation of the stopping power for heavy ions with incident energies
above 1 MeV/amu.
Nuclear stopping for incident heavy ions is calculated with the universal potential from
ref. [16]. The reduced nuclear stopping sn with the universal potential is given by
sn =
ln(1 + 1.1383 ²)
2 [² + 0.01321 ²0.21226 + 0.19593 ²0.5 ]
(4.26)
for ² ≤ 30. For ² > 30 sn is given by
sn =
ln(²)
2²
(4.27)
The reduced energy ² in eqs. 4.26 and 4.27 is calculated using the universal screening
length aU , which is ∝ 1/(Z10.23 + Z20.23 ) instead of the Firsov screening length aF ∝
2/3
1/(Z1
2/3
+ Z2 )1/2 , which is used in eq. 4.25. The difference between eq. 4.24 and 4.26 is
only some percent.
The nuclear stopping component is only important at ion energies below about
200 keV/amu. Nuclear stopping becomes very small at higher energies, typically below
1% of the electronic stopping component.
4.4.4
Stopping in compounds
SIMNRA uses Bragg’s rule [17] for the determination of the stopping power in compounds.
Bragg’s rule is a simple linear additivity rule of the stopping contributions of the different
compound elements, assuming that the interaction of an incident ion with a target atom
is independent of the surrounding target atoms. For a compound consisting of different
P
elements i with atomic concentrations ci (
by
S=
ci = 1) the total stopping power S is given
X
42
ci Si
(4.28)
Si is the stopping power of each element.
Bragg’s rule assumes that the interaction between the ion and the atom is independent
of the environment. The chemical and physical state of the medium is, however, observed
to have an effect on the energy loss. The deviations from Bragg’s rule predictions are most
pronounced around the stopping power maximum and for solid compounds containing
heavier constituents, such as oxides, nitrides, hydrocarbons, etc. The deviations from
Bragg’s rule predictions may be of the order of 10–20% [18].
Ziegler and Manoyan [19] have developed the ’cores and bonds’ (CAB) model, which
assumes the electronic energy loss to have two contributions: The effect of the cores and
the effect of the bonds, such as C-H and C-C. The CAB-model allows better predictions
for the stopping in compounds, however, the bond structure has to be known. Currently
the CAB-model (or any other model which allows better predictions for the stopping in
compounds) is not implemented in SIMNRA.
4.5
Straggling
When a beam of charged particles penetrates matter, the slowing down is accompanied by
a spread in the beam energy. This phenomenon is called straggling. It is due to statistical
fluctuations of the energy transfer in the collision processes.
Energy loss straggling has different contributions:
1. Electronic energy loss straggling due to statistical fluctuations in the transfer of
energy to electrons.
2. Nuclear energy loss straggling due to statistical fluctuations in the nuclear energy
loss. Though the contribution of the nuclear energy loss to the total energy loss is
usually negligible at high energies, it contributes to the total energy loss straggling.
The contribution of the nuclear energy loss straggling is, however, always smaller
than the contribution of the electronic energy loss straggling.
3. Straggling due to plural scattering and multiple small angle scattering, resulting in
different path lengths and energy and angular spread of the incident beam.
4. Geometrical straggling due to finite detector solid angle and finite beam spot size,
resulting in a distribution of scattering angles and different pathlengths for outgoing
43
particles.
5. Straggling due to surface and interlayer roughness.
An additional contribution to the energy broadening visible in experimental spectra
is the energy resolution of the detector. The different straggling contributions have been
recently reviewed by Szil´agy et. al. [20].
Multiple small angle scattering, geometrical straggling and surface roughness are not
calculated by SIMNRA. The finite energy resolution of the detector is included in the
calculations. Plural scattering with two scattering events (= dual scattering) can be
calculated by SIMNRA.
44
4.5.1
Electronic and nuclear energy loss straggling
There are four main theories describing electronic and nuclear energy loss straggling [21,
22, 23], each applicable in a different regime of energy loss. With ∆E the mean energy
loss of the beam, and E the energy of the incident beam, we can distinguish:
∆E/E < 10%
Vavilov’s Theory[24, 22]. For thin layers and small energy losses.
The energy distribution is non-Gaussian and asymmetrical. This
energy range is not described properly by SIMNRA.
10 − 20%
Bohr’s Theory[25, 26]. As the number of collisions becomes large,
the distribution of particle energies becomes Gaussian.
20 − 50%
Symon’s Theory[21]. This theory includes non-statistical broadening caused by the change in stopping power over the particle energy
distribution. If the mean energy of the beam is higher than the energy
of the stopping power maximum, then particles with a lower energy
have a higher stopping power, and particles with higher energy have
a smaller stopping power. This results in a nonstatistical broadening
of the energy distribution. The width of the particles energy distribution in Symon’s theory is significantly higher than predicted by
Bohr’s theory. The distribution of particle energies is still Gaussian.
50 − 90%
Payne’s and Tschal¨
ars Theory [27, 28, 29]. When the energy
losses become very large and the mean energy of the beam decreases
below the energy of the stopping power maximum, the particle energy
distribution again become skewed, because now particles with lower
energy have a lower stopping power than particles with higher energy.
The distribution is about Gaussian.
SIMNRA always assumes that the particles energy distribution is Gaussian. This is
only an approximation for thin layers: In this case the energy distribution is described
by the Vavilov distribution [24, 22]. However, the straggling contribution of thin layers
to the total energy broadening is much smaller than the contribution of the finite energy
resolution of the detector.
SIMNRA calculates the non-statistic broadening (or skewing) of the energy distribution
in the following way:
45
Assume two particles with energies E1 and E2
∆E
2
∆E
= E0 −
2
E1 = E0 +
E2
centered around a mean energy E0 . The energy difference E1 − E2 of the two particles
is ∆E. To evaluate the stopping power ² = dE/dx at E1 and E2 we can apply a Taylor
expansion of the stopping power ²(E) and considering only the linear term:
1 d²
(E0 )∆E
2 dE
1 d²
²(E2 ) = ²(E0 ) −
(E0 )∆E.
2 dE
²(E1 ) = ²(E0 ) +
The energies E10 and E20 after penetrating a thin layer with thickness ∆x are then given
by
E10 = E1 − ²(E1 )∆x
E20 = E2 − ²(E2 )∆x.
By putting the above equations together we get the energy difference ∆E 0 = E10 − E20 of
the two particles after penetrating the layer:
¶
µ
∆E 0 = 1 −
d²
(E0 )∆x ∆E.
dE
(4.29)
If d²/dE is negative, which is the case for all energies above the stopping power maximum,
the energy difference increases and the distribution function is broadened. If d²/dE is
positive, the energy difference decreases and the distribution function gets skewed. Because
this is a linear Taylor expansion, a Gaussian distribution remains gaussian, but the width
of the Gaussian is changed according to eq. 4.29.
To the non-statistic broadening we have to add the statistical effects. When the incident beam with initial energy E0 and initial beam width σ02 (σ 2 is the variance of the
√
energy distribution, the full width at half maximum (FWHM) is 2 2 ln 2 σ = 2.355 σ)
penetrates a layer of matter with thickness ∆x, then the beam width σ12 after penetrating
the layer is given by:
µ
σ12 = 1 −
d²
(E0 )∆x
dE
¶2
σ02 + σ 2
(4.30)
²(E) = dE/dx(E) is the stopping power of the material and σ 2 is the energy loss straggling
in the layer including nuclear energy loss straggling in Bohr approximation and electronic
46
energy loss straggling according to Chu’s theory. The first term in eq. 4.30 describes the
non-statistical broadening of the beam according to eq. 4.29 due to the energy dependence
of the stopping power, the second term adds the statistical effects.
The variance of the energy loss straggling σ 2 in eq. 4.30 has two contributions: nuclear
energy loss straggling σn2 and electronic energy loss straggling σe2 . The total straggling
σ 2 is given by quadratically adding the two independent contributions of electronic and
nuclear straggling:
σ 2 = σe2 + σn2
(4.31)
The electronic energy loss straggling is calculated by applying Chu’s theory [30, 26]:
2
σe2 = H(E/M1 , Z2 )σBohr
(4.32)
2
σBohr
is the electronic energy loss straggling in Bohr approximation and is given by [25, 26]:
2
σBohr
[keV2 ] = 0.26 Z12 Z2 ∆x [1018 atoms/cm2 ]
(4.33)
Bohr’s theory of electronic energy loss straggling is valid in the limit of high ion velocities.
In this case the electronic energy loss straggling is almost independent of the ion energy.
For lower ion energies the Bohr straggling is multiplied by the Chu correction factor
H(E/M1 , Z2 ), which depends only on E/M1 and the nuclear charge of the target atoms
Z2 . H takes into account the deviations from Bohr straggling caused by the electron
binding in the target atoms. Chu [30, 26] has calculated H by using the Hartree-FockSlater charge distribution. This calculation gives straggling values which are considerably
lower than those given by Bohr’s theory. The correction factor H, as used by SIMNRA,
is shown in fig. 4.3. The Z2 oscillations are clearly visible. The Chu correction is mainly
necessary for high Z2 and low energies. For high energies H approaches 1 and becomes
independent of Z2 and energy. For E/M1 values in the range 100–1000 keV/amu the
data have been taken from ref. [26], data for lower and higher E/M1 values are based
on an extrapolation performed in ref. [20]. Tabulated values for H are stored in the file
CHU CORR.DAT. For not tabulated values SIMNRA uses linear interpolation.
For the nuclear energy loss straggling σn2 SIMNRA uses Bohr’s theory of nuclear straggling. The nuclear energy loss straggling in Bohr’s approximation is given by:
µ
σn2
2
[keV ] =
0.26 Z12 Z22
M1
M1 + M2
47
¶2
∆x [1018 atoms/cm2 ]
(4.34)
1.2
E/M
[keV/amu]
1.0
5000
H(E/M, Z2)
0.8
3000
2000
1500
0.6
1000
750
500
300
200
100
50
10
0.4
0.2
0.0
0
20
40
60
80
100
Z2
Figure 4.3: The Chu straggling correction for several values of E/M1 as a function of the nuclear
charge of the target Z2 . Dots are original data from Chu [26], solid lines are extrapolated data
taken from [20].
48
4
2.5 MeV He in Si
Straggling (keV FWHM)
80
SIMNRA
Bohr
60
40
20
10%
20%
Max
50%
0
0
1
2
Depth (10
3
19
4
2
atoms/cm )
Figure 4.4: Beam width (FWHM) of 2.5 MeV 4 He ions penetrating through silicon. The solid
line is the beam width calculated by SIMNRA using eq. 4.30, the dashed line is the prediction of
Bohr’s theory. The vertical lines denote the mean depth at which the beam has lost 10%, 20%
and 50% of its initial energy. Max denotes the depth at which the mean energy of the beam has
decreased to the energy of the stopping power maximum.
Because the nuclear energy loss straggling is generally much smaller than the electronic
energy loss straggling, Bohr’s theory of nuclear straggling can be used without loss in
precision.
Fig. 4.4 compares the beam width (FWHM) of 2.5 MeV 4 He ions in silicon calculated
by SIMNRA using eq. 4.30 with Bohr’s theory. For small energy losses the beam width
calculated by SIMNRA is slightly smaller than predicted by Bohr’s theory due to the Chu
correction. However, this is counterbalanced by the nonstochastic broadening due to the
characteristics of the stopping power curve, and for larger energy losses the beam width
gets larger than in Bohr’s theory. When the mean beam energy has decreased below the
energy of the stopping power maximum, the beam width becomes skewed.
As can be seen from fig. 4.3 the deviation of the Chu correction from Bohr’s theory is
largest for high Z2 and low energies. Fig. 4.5 shows the beam width (FWHM) of 1 MeV
4 He
penetrating through gold. The deviation from Bohr’s theory is large. The stopping
power maximum is at about 960 keV. For low energy losses the beam width increases due
49
4
1.0 MeV He in Au
Straggling (keV FWHM)
80
SIMNRA
Bohr
60
40
Max
20%
50%
90%
20
0
0
2
4
6
Depth (10
18
8
10
2
atoms/cm )
Figure 4.5: Beam width (FWHM) of 1.0 MeV 4 He ions penetrating through gold. The solid line
is the beam width calculated by SIMNRA using eq. 4.30, the dashed line is the prediction of Bohr’s
theory. The vertical lines denote the mean depth at which the beam has lost 20%, 50% and 90%
of its initial energy. Max denotes the depth at which the mean energy of the beam has decreased
to the energy of the stopping power maximum.
to the statistical broadening because the nonstochastic skewing, which occurs for beam
energies below the stopping power maximum, is small and the stochastic broadening wins.
For larger energy losses however the beam width gets skewed.
Fig. 4.6 shows the measured RBS-spectrum for 1.0 MeV 4 He ions incident on a gold
layer with a thickness of about 100 nm and a scattering angle of 165◦ compared with simulations using Bohr straggling and Chu straggling. As can be seen at the low energy edge of
the layer, the Bohr straggling is broader than the experimental data. The Chu straggling
fits the measured curve relatively well, except of the multiple scattering contribution.
The straggling of outgoing particles is calculated with eq. 4.30 as well. Outgoing
2
particles always start with an energy distribution with variance σout
which is given by
2
2
σout
= K 2 σin
(4.35)
2 the variance of the energy distribution of the incident
with K the kinematic factor and σin
beam.
50
Energy (keV)
700
750
800
850
900
950
Experimental
Chu
Bohr
6000
Counts
1000
4000
2000
0
500
600
700
Channel
Energy (keV)
740
750
760
770
780
790
800
6000
4000
2000
0
540
560
580
600
Channel
Figure 4.6: Measured and simulated spectra using Bohr and Chu straggling of 1.0 MeV 4 He ions
incident on 100 nm Au on Si, scattering angle 165◦ . Bottom: Magnification of the low energy gold
edge.
51
Figure 4.7: Examples of ion trajectories with one, two and three scattering events.
4.5.2
Energy loss straggling in compounds
For compounds a simple additivity rule for energy loss straggling is used [26]. The straggling in a compound consisting of elements i with atomic concentration ci is calculated
with
σ2 =
X
ci σi2
(4.36)
i
with σi2 being the straggling in each element.
4.6
Multiple and plural scattering
SIMNRA uses straight lines as trajectories for the ingoing and outgoing particles, with
one single scattering event connecting the trajectories of the particles, see fig. 4.7 left.
This is only an approximation to physical reality, because the particles on the ingoing and
outgoing path suffer many small angle scatterings with small scattering angles (this has
been called multiple scattering) and additionally may perform more than one scattering
event with large scattering angle (see fig. 4.7 middle and right), before they are scattered
towards the detector. This has been called plural scattering.
Multiple scattering has been recently reviewed by Szilagy et al. [20]. Multiple scattering results in an angular spread of the particles and therefore in a spread of path lengths.
Due to the path length differences, we get an energy spread of the particles in a given
depth. Multiple scattering is not treated by SIMNRA.
Plural scattering with 2, 3, 4, . . . scattering events is responsible for the background
behind the low energy edge of high Z elements on top of low Z elements and the steeper
increase of the spectra towards low energies than calculated with single scattering [31,
32]. SIMNRA is able to calculate dual scattering, i. e. all particle trajectories with two
scattering events.
52
SIMNRA performs the calculation of dual scattering in the following way: During each
step of the incident ion particles are scattered into the whole sphere of 4π. We introduce
the polar system with the polar angles ψ, φ, see fig. 4.8. After the first scattering event the
scattered particles have the direction ψ, φ. The scattering angle θ1 of the first scattering
event is given by
cos θ1 = sin α sin ψ sin φ − cos α cos ψ
SIMNRA uses the Rutherford cross-section for the calculation of the number of scattered
particles in the first scattering event. The new angle α0 of the particles after the first
scattering is
α0 = 180◦ − ψ
The scattering angle θ2 of the second scattering event is given by
cos θ2 = sin β sin ψ sin φ + cos β cos ψ
SIMNRA subdivides the whole sphere of 4π into 10 ψ-intervals and 12 φ-intervals, resulting
in 120 solid angle intervals. SIMNRA considers only trajectories with scattering angles
θ1 , θ2 > 20◦ for dual scattering. Trajectories with smaller scattering angles are very similar
to single scattering trajectories. For each solid angle interval a full backscattering spectrum
with the starting depth of the particles equal to the depth of the incident ions and the
new incident angle α0 and the new scattering angle θ2 is calculated.
Fig. 4.9 compares the simulated spectra with single and dual scattering for 500 keV
4 He
ions incident on a 100 nm gold layer on top of silicon with experimental data. With
the inclusion of dual scattering the experimental results are much better approximated.
Dual scattering gives the background between the low energy edge of Au and the Si edge,
and the steeper increase of the gold spectrum is better described. The results with dual
scattering are slightly lower than the experimental results. This is due to trajectories with
more than two scattering events.
53
z
ψ
x
ϕ
y
α
β
Figure 4.8: Geometry used for the calculation of dual scattering.
Energy (keV)
100
14000
12000
200
300
Experimental
Dual scattering
Single scattering
400
500
Au
Counts
10000
8000
6000
4000
Si
2000
0
100
200
300
Channel
Figure 4.9: 500 keV 4 He ions incident on 100 nm Au on top of Si, scattering angle 165◦ . Circles:
experimental data points, dashed line: simulation with one scattering event, solid line: simulation
with two scattering events.
54
Chapter 5
Examples
This chapter gives several examples for the abilities of SIMNRA.
All backscattering spectra were measured at the IPP Garching at a scattering angle
θ = 165◦ . The solid angle of the detector was 1.08 × 10−3 sr. A standard surface barrier
detector with a nominal energy resolution of 15 keV FWHM was used.
5.1
RBS: Rutherford cross-sections
Fig. 5.1 shows the measured and simulated spectra for 1.0 MeV 4 He incident ions on a
gold layer with a thickness of about 100 nm on top of silicon. The simulated spectrum
fits the measured data very well. The low background between the Si edge and the low
energy Au edge is due to plural scattering (this means the backscattered particles have
suffered more than one scattering event with large scattering angle) [31, 32], which was
not simulated for this example. The deviation between experiment and simulation at low
energies in the Si spectrum is due to the same reason.
Fig. 5.2 compares simulated spectra with single and dual scattering for 500 keV 4 He
ions incident on a 100 nm gold layer on top of silicon with experimental data. At this
low energy plural scattering is important. With the inclusion of dual scattering the experimental results are much better approximated. Dual scattering gives the background
between the low energy edge of Au and the Si edge, and the steeper increase of the gold
spectrum is better described. The results with dual scattering are slightly lower than the
experimental results. This is due to trajectories with more than two scattering events,
which are not calculated.
55
Energy (keV)
200
600
800
experimental
simulated
6000
Counts
400
1000
Au
4000
2000
Si
0
100
200
300
400
500
600
700
800
Channel
Figure 5.1: 1000 keV 4 He incident on Au on top of silicon, θ = 165◦ .
56
Energy (keV)
100
14000
12000
200
300
Experimental
Dual scattering
Single scattering
400
500
Au
Counts
10000
8000
6000
4000
Si
2000
0
100
200
300
Channel
Figure 5.2: 500 keV 4 He ions incident on 100 nm Au on top of Si, scattering angle 165◦ . Circles:
experimental data points, dashed line: simulation with one scattering event, solid line: simulation
with two scattering events.
57
Energy (keV)
400
600
800
1000
1200
14000
1400
experimental
simulated
12000
Counts
10000
8000
6000
4000
2000
0
100
200
300
400
500
600
Channel
Figure 5.3: 2000 keV protons on carbon (HOPG), α = 5◦ , θ = 165◦ .
5.2
RBS: Non-Rutherford cross-sections
Fig. 5.3 shows the measured and simulated spectra for 2.0 MeV protons incident on highly
oriented pyrolytic graphite (HOPG). To avoid channelling the incident angle α was 5◦ .
The cross-section is non-Rutherford, and the cross-section data of Amirikas et. al. [33]
were used for the simulation. The pronounced peak in the spectrum is due to the resonance
in the
12 C(p,p)12 C
cross-section at 1732 keV. The measured and simulated spectra agree
very well.
Fig. 5.4 shows the measured and simulated spectra for 2.0 MeV protons incident on
silicon. To avoid channelling the incident angle α was 5◦ . The cross-section is nonRutherford, and the cross-section data of Vorona et. al. [34] were used for the simulation.
As in the case of carbon the measured and simulated spectra agree very well. The structures in the simulated spectrum between channel 500 and 700 are due to the experimentally
determined cross-section data, which contain these structures.
58
Energy (keV)
400
600
800
1000 1200 1400 1600 1800
experimental
simulated
Counts
2000
0
100
200
300
400
500
600
700
800
Channel
Figure 5.4: 2000 keV protons backscattered from silicon, α = 5◦ , θ = 165◦ .
59
Energy (keV)
600
800
1000
1200
1400
1600
1800
Exp. SIMNRA
200
H
D
Counts
150
100
50
0
100
200
300
Channel
Figure 5.5: ERDA with 2.6 MeV 4 He ions incident on a soft amorphous hydrocarbon layer (a:CH layer) containing both H and D. The recoiling H and D atoms were separated with a ∆E-E
telescope detector, the backscattered 4 He ions are not shown. α = β = 75◦ , θ = 30◦ .
5.3
ERDA: Non-Rutherford cross-sections
Fig 5.5 shows the measured and simulated spectra for ERDA with 2.6 MeV incident 4 He
ions on a soft amorphous hydrocarbon layer (a:C-H layer) containing both H and D.
The recoiling H and D atoms were separated with a ∆E-E telescope detector [35]. Both
recoil cross-sections are non-Rutherford. The cross-section data of Baglin et al. [36] for
H(4 He,H)4 He and of Besenbacher et al. [37] for D(4 He,D)4 He were used for the simulation.
The peak in the deuterium spectrum is due to a resonance at a 4 He energy of 2130 keV.
The measured and simulated data agree very well.
60
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